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Abstract:

A method for making downhole whirl measurements in a drill string
includes rotating a sensor set in a borehole. The sensor set is deployed
in the drill string and includes at least one cross-axial accelerometer
and at least one cross-axial magnetometer. Sensor measurements, including
a plurality of accelerometer measurements and a plurality of magnetometer
measurements made at predetermined measurement intervals, may be obtained
while drilling and used to compute a whirl magnitude.

Claims:

1. A method for making downhole whirl measurements in a drill string, the
method comprising: (a) rotating a sensor set in a borehole, the sensor
set deployed in the drill string and including at least one cross-axial
accelerometer and at least one cross-axial magnetometer; (b) causing the
sensor set to obtain sensor measurements, the sensor measurements
including a plurality of accelerometer measurements and a plurality of
magnetometer measurements at predetermined measurement intervals; and (c)
processing the sensor measurements obtained in (b) and a predetermined
blade count to obtain a whirl magnitude.

2. The method of claim 1, wherein (c) comprises: (i) processing the
sensor measurements obtained in (b) to obtain a rotation rate of the
drill string; (ii) processing the rotation rate obtained in (c) and a
stabilizer or drill bit blade count to obtain a whirl frequency; and
(iii) processing the sensor measurements obtained in (b) and the whirl
frequency obtained in (ii) to obtain the whirl magnitude.

3. The method of claim 2, wherein said processing in (iii) further
comprises applying a digital band pass filter to the accelerometer
measurements obtained in (b), the band pass filter having a center
frequency substantially equal to the whirl frequency obtained in (ii).

4. The method of claim 2, wherein said processing in (iii) further
comprises transforming the accelerometer measurements obtained in (b) to
a frequency domain and obtaining a signal amplitude of said transformed
accelerometer measurements at a frequency substantially equal to the
whirl frequency obtained in (ii).

5. The method of claim 2, wherein said processing in (i) further
comprises differentiating the magnetometer measurements obtained in (b)
to obtain the rotation rate of the drill string.

6. The method of claim 1, further comprising: (d) transmitting the whirl
magnitude to a surface location; and (e) processing the whirl magnitude
at the surface location to automatically adjust at least one drilling
parameter selected from the group consisting of weight on bit, drill
string rotation rate, and drilling fluid flow rate.

7. A method for making downhole bit whirl measurements, the method
comprising: (a) rotating a navigational sensor set in a borehole, the
navigational sensor set deployed in the drill string and including at
least a tri-axial accelerometer set and a tri-axial magnetometer set; (b)
causing the navigational sensor set to obtain sensor measurements, the
sensor measurements including a plurality of accelerometer measurements
and a plurality of magnetometer measurements at predetermined measurement
intervals; and (c) processing the sensor measurements obtained in (b) and
a predetermined drill bit blade count to obtain a bit whirl magnitude.

8. The method of claim 7, wherein (c) comprises: (i) processing the
sensor measurements obtained in (b) to obtain a rotation rate of the
drill string; (ii) processing the rotation rate obtained in (c) and a
drill bit blade count to obtain a bit whirl frequency; and (iii)
processing the sensor measurements obtained in (b) and the bit whirl
frequency obtained in (ii) to obtain the bit whirl magnitude.

9. The method of claim 8, wherein said processing in (iii) further
comprises applying a digital band pass filter to the accelerometer
measurements obtained in (b), the band pass filter having a center
frequency substantially equal to the bit whirl frequency obtained in
(ii).

10. The method of claim 9, wherein the digital band pass filter is
computed downhole based upon the bit whirl frequency obtained in (ii).

11. The method of claim 9, wherein the digital band pass filter is
selected from tool memory based upon the bit whirl frequency obtained in
(ii).

12. The method of claim 8, wherein said processing in (iii) further
comprises transforming the accelerometer measurements obtained in (b) to
a frequency domain and obtaining a signal amplitude of said transformed
accelerometer measurements at a frequency substantially equal to the bit
whirl frequency obtained in (ii).

13. The method of claim 12, wherein the accelerometer measurements are
transformed to the frequency domain using a Fast Fourier Transform.

14. The method of claim 8, wherein said processing in (i) further
comprises differentiating the magnetometer measurements obtained in (b)
to obtain the rotation rate of the drill string.

15. The method of claim 8, wherein bit whirl frequency is computed in
(ii) according to the following equation: ωwhirl=ω(N+1)
wherein ωwhirl represents the bit whirl frequency, ω
represents a rotational frequency of the drill string obtained in (i),
and N represents a number of blades on a PDC drill bit.

16. The method of claim 7, wherein the navigational sensor set is
deployed within two meters of a drill bit.

17. The method of claim 7, wherein the predetermine measurement interval
is in the range from about 0.0001 to about 0.1 second.

18. The method of claim 7, further comprising: (d) transmitting the bit
whirl magnitude to a surface location.

19. The method of claim 18, further comprising: (e) processing the bit
whirl magnitude at the surface location to automatically adjust at least
one drilling parameter selected from the group consisting of weight on
bit, drill string rotation rate, and drilling fluid flow rate.

20. The method of claim 19, wherein (e) further comprises processing a
stick/slip measurement to automatically adjust at least one of the
drilling parameters.

[0002] Disclosed embodiments relate generally to dynamics measurements
made while drilling and more particularly to a method for detecting and
quantifying bit whirl while drilling.

BACKGROUND

[0003] It is well known in the art that severe dynamic conditions are
sometimes encountered during drilling. Commonly encountered dynamic
conditions include, for example, axial vibration, lateral shock and
vibration, torsional vibration, stick/slip, and whirl. Bit bounce
includes axial vibration of the drill string, sometimes resulting in
temporary lift off of the drill bit from the formation ("bouncing" of the
drill bit off the bottom of the borehole). Axial vibrations (e.g., bit
bounce) is known to reduce the rate of penetration (ROP) during drilling,
may cause excessive fatigue damage to BHA components, and may even damage
the well in extreme conditions.

[0004] Lateral vibrations are those which are transverse to the axis of
the drill string (cross-axial). Such lateral vibrations are commonly
recognized as a leading cause of drill string, drill string connection,
and BHA failures and may be caused, for example, by bit whirl and/or the
use of unbalanced drill string components.

[0005] Stick/slip refers to a torsional vibration induced by friction
between drill string components and the borehole wall. Stick/slip is
known to produce instantaneous drill string rotation speeds many times
that of the nominal rotation speed of the table. In stick/slip conditions
a portion of the drill string or bit sticks to the borehole wall due to
frictional forces often causing the drill string to temporarily stop
rotating. Meanwhile, the rotary table continues to turn resulting in an
accumulation of torsional energy in the drill string. When the torsional
energy exceeds the static friction between the drill string and the
borehole, the energy is released suddenly in a rapid burst of drill
string rotation. Instantaneous downhole rotation rotates have been
reported to exceed four to ten times that of the rotary table. Stick/slip
is known to cause severe damage to downhole tools, as well as connection
fatigue, and excess wear to the drill bit and near-bit stabilizer blades.
Such wear commonly results in reduced ROP and loss of steerability in
deviated boreholes.

[0006] Bit or stabilizer whirl may be caused by the instantaneous center
of rotation moving around the face of the bit (or about the axis of the
string). The movement (rotation of the whirl) is generally in the
opposite direction of the rotation of the drill string (counterclockwise
vs. clockwise). Cutting elements on a whirling bit have been documented
to move sideways, backwards, and at much higher velocities than those on
a non-whirling bit. The associated impact loads are known to cause
chipping and accelerated wear of the bit components. For example, severe
bit damage has been observed even after very short duration drilling
operations for polycrystalline diamond compact (PDC) bits.

[0007] These harmful dynamic conditions not only cause premature failure
and excessive wear of the drilling components, but also can result in
costly trips (tripping-in and tripping-out of the borehole) due to
unexpected tool failures and wear. Furthermore, there is a trend in the
industry towards drilling deeper, smaller diameter wells where damaging
dynamic conditions can become increasingly problematic. Problems
associated with premature tool failure and wear are exacerbated (and more
expensive) in such wells.

[0008] The above-described downhole dynamic conditions are known to be
influenced by drilling parameters. By controlling such drilling
parameters an operator can sometimes mitigate against damaging dynamic
conditions. For example, bit bounce and lateral vibration conditions can
sometimes be overcome by reducing both the weight on bit and the drill
string rotation rate. Stick/slip conditions can often be overcome via
increasing the drill string rotation rate and reducing weight on bit. The
use of less aggressive drill bits also tends to reduce bit bounce,
lateral vibrations, and stick/slip in many types of formations. The use
of stiffer drill string components is further known to sometimes reduce
stick/slip. While the damaging dynamic conditions may often be mitigated
as described above, reliable measurement and identification of such
dynamic conditions can be problematic. For example, lateral vibration and
stick/slip conditions are not easily quantified by surface measurements.
In fact, such dynamic conditions are sometimes not even detectable at the
surface, especially at vibration frequencies above about 5 hertz.

[0009] Downhole dynamics measurement systems have been known in the art
for at least 15 years. While these, and other known systems and methods,
may be serviceable in certain applications, there is yet need for further
improvement. For example, known systems typically make use of dedicated
sensors which tends to increase costs and expend valuable BHA real estate
(e.g., via the introduction of a dedicated dynamics measurement sub).
Also, such dedicated sensors tend to increase power consumption and
component counts and, therefore, the complexity of MWD, LWD, and
directional drilling tools, and thus tend to reduce reliability of the
system. Moreover, dedicated sensors must typically be deployed a
significant distance above the drill bit.

[0010] Therefore there exists a need for an improved method for making
downhole dynamics measurements and particularly for making such
measurements as close to the drill bit as possible.

SUMMARY

[0011] A method for making downhole whirl measurements (such as bit whirl
measurements) in a drill string is disclosed. The method includes
rotating a downhole sensor set in a borehole. The sensor set is deployed
in the drill string and includes at least first and second cross-axial
accelerometer and first and second cross-axial magnetometers. The sensors
used to obtain sensor measurements including a plurality of accelerometer
measurements and a plurality of magnetometer measurements at
predetermined measurement intervals. The sensor measurements are then
processed in combination with a predetermined blade count to obtain a
whirl magnitude. The sensor measurements may be processed to obtain a
rotation rate of the drill string, which may be in turn further processed
in combination with a stabilizer or drill bit blade count to obtain a
whirl frequency. The sensor measurements may then be processed in
combination with the whirl frequency to obtain the whirl magnitude.

[0012] The disclosed embodiments may provide various technical advantages.
For example, the disclosed method may make use of existing accelerometer
and magnetometer sensor sets to obtain whirl frequency and magnitude
parameters while drilling. Moreover, the sensors may be deployed very
close to the drill bit enabling the acquisition of bit whirl parameters
(such as bit whirl magnitude). Measuring the bit whirl magnitude while
drilling may enable an operator to prevent damage to the drill string.
For example, drill string rotation rate and weight on bit may be
controlled at the surface in a closed loop fashion to automatically
mitigate harmful vibrations.

[0013] This summary is provided to introduce a selection of concepts that
are further described below in the detailed description. This summary is
not intended to identify key or essential features of the claimed subject
matter, nor is it intended to be used as an aid in limiting the scope of
the claimed subject matter.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] For a more complete understanding of the disclosed subject matter,
and advantages thereof, reference is now made to the following
descriptions taken in conjunction with the accompanying drawings, in
which:

[0015] FIG. 1 depicts one example of a conventional drilling rig on which
disclosed methods may be utilized.

[0023] FIG. 1 depicts an example drilling rig 10 suitable for using
various method embodiments disclosed herein. A semisubmersible drilling
platform 12 is positioned over an oil or gas formation (not shown)
disposed below the sea floor 16. A subsea conduit 18 extends from deck 20
of platform 12 to a wellhead installation 22. The platform may include a
derrick and a hoisting apparatus for raising and lowering a drill string
30, which, as shown, extends into borehole 40 and includes a drill bit 32
and a near-bit sensor sub 60 (such as the iPZIG® tool available from
PathFinder®, A Schlumberger Company, Katy, Tex., USA). Drill string
30 may further include a downhole drilling motor, a steering tool such as
a rotary steerable tool, a downhole telemetry system, and one or more MWD
or LWD tools including various sensors for sensing downhole
characteristics of the borehole and the surrounding formation. The
disclosed embodiments are not limited in these regards.

[0024] It will be understood by those of ordinary skill in the art that
the deployment illustrated on FIG. 1 is merely an example. It will be
further understood that disclosed embodiments are not limited to use with
a semisubmersible platform 12 as illustrated on FIG. 1. The disclosed
embodiments are equally well suited for use with any kind of subterranean
drilling operation, either offshore or onshore.

[0025] FIG. 2 depicts the lower BHA portion of drill string 30 including
drill bit 32, a near-bit sensor sub 60, and a lower portion of a steering
tool 80. In the depicted embodiment, sensor sub body 62 is threadably
connected with the drill bit 32 and therefore configured to rotate with
the drill bit 32. The sensor sub 60 includes tri-axial accelerometer 65
and magnetometer 67 navigation sensors and may optionally further include
a logging while drilling sensor 70 such as a natural gamma ray sensor. In
the depicted embodiment, the sensors 65 and 67 may be deployed as closed
to the bit 32 as possible, for example, within two meters, or even within
one meter, of the bit 32.

[0026] Suitable accelerometers for use in sensors 65 and 67 may be chosen
from among substantially any suitable commercially available devices
known in the art. For example, suitable accelerometers may include Part
Number 979-0273-001 commercially available from Honeywell, and Part
Number JA-5H175-1 commercially available from Japan Aviation Electronics
Industry, Ltd. (JAE). Suitable accelerometers may alternatively include
micro-electro-mechanical systems (MEMS) solid-state accelerometers,
available, for example, from Analog Devices, Inc. (Norwood, Mass.). Such
MEMS accelerometers may be advantageous for certain near bit sensor sub
applications since they tend to be shock resistant, high-temperature
rated, and inexpensive. Suitable magnetic field sensors may include
conventional three-axis ring core flux gate magnetometers or conventional
magnetoresistive sensors, for example, Part Number HMC-1021D, available
from Honeywell.

[0027] FIG. 2 further includes a diagrammatic representation of the
tri-axial accelerometer and magnetometer sensor sets 65 and 67. By
tri-axial it is meant that each sensor set includes three mutually
perpendicular sensors, the accelerometers being designated as Ax,
Ay, and Az and the magnetometers being designated as BX,
By, and B. By convention, the z-axis accelerometer and magnetometer
(Az and Bz) are oriented substantially parallel with the
borehole as indicated (although disclosed embodiments are not limited in
this regard). Each of the accelerometer and magnetometer sets may
therefore be considered as determining a plane (the x and y-axes) and a
pole (the z-axis along the axis of the BHA).

[0028] The accelerometer and magnetometer sets may be configured for
making downhole navigational (surveying) measurements during a drilling
operation. Such measurements are well known and commonly used to
determine, for example, borehole inclination, borehole azimuth, gravity
toolface, and magnetic toolface. Being configured for making navigational
measurements, the accelerometer and magnetometer sets 65 and 67 are
rotationally coupled to the drill bit 32 (e.g., rotationally fixed to the
sub body 62 which rotates with the drill bit). The accelerometers may
also be electronically coupled to a digital controller via a low-pass
filter (including an anti-aliasing filter) arrangement. Such "DC
coupling" is generally desirable for making accelerometer based surveying
measurements (e.g., borehole inclination or gravity toolface
measurements). The use of a low-pass filter band-limits sensor noise
(including noise caused by sensor vibration) and therefore tends to
improve sensor resolution and surveying accuracy.

[0029] FIG. 3 depicts a circular cross sectional view of one example
accelerometer arrangement in sensor sub 60. In the depicted embodiment,
the x-axis and y-axis accelerometers 65x and 65y are circumferentially
spaced apart from one another by about 90 degrees. The z-axis
accelerometer is not depicted and may be deployed substantially anywhere
in the sub body 62. The accelerometers 65x and 65y may each be aligned
with a radial direction 69 such that each accelerometer is substantially
insensitive to centripetal acceleration (i.e., the radially directed
acceleration caused by a uniform rotation of the sub body 62). The
accelerometers remain sensitive to tangential acceleration (i.e.,
acceleration caused by non-uniform rotation of the sub body 62). The
arrangement therefore remains sensitive to stick/slip (torsional
vibration) conditions. It will be understood that the disclosed method
embodiments are not limited to use with the depicted accelerometer
arrangement. For example, accelerometers 65x and 65y may be deployed at
substantially the same location in the tool body 62. The accelerometers
65x and 65y may alternatively be aligned with a tangential direction such
that they are substantially insensitive to tangential acceleration and
sensitive to centripetal (radial) acceleration.

[0030] FIG. 4 depicts a flow chart of one example of a method 100 for
making downhole dynamics measurements with rotating navigational sensors.
Navigational sensors are rotated in a borehole at 102, for example, while
drilling the borehole (by either rotating the drill string at the surface
or rotating the drill bit with a conventional mud motor). Conventionally,
the x- and y-axis navigation sensor data are unused while the sensors are
rotated (e.g., drill string or drill bit rotation during drilling). The
navigational sensors may include a tri-axial accelerometer set and a
tri-axial magnetometer set as described above with respect to FIGS. 2 and
3. Moreover, the sensors may be deployed as close to the bit as possible,
for example, in a near-bit sensor sub as is also described above with
respect to FIGS. 2 and 3.

[0031] Accelerometer measurements are made at a predetermined time
interval at 104 while rotating in 102 (e.g., during the actual drilling
process) to obtain a set (or array) of accelerometer measurements. The
accelerometer measurements may then be digitally (numerically)
differentiated at 106 to remove a DC component of the acceleration and
obtain a set of differentiated accelerometer measurements (i.e.,
acceleration differences). Maximum and minimum difference values obtained
over some time period or number of difference samples may then be
processed at 108 to obtain a drill string vibration parameter. This
process may be optionally repeated substantially any number of times at
110 to obtain an averaged difference value at 112. This averaged value
may then be taken as an indication of lateral or axial vibration as is
described in more detail below.

[0032] It will be appreciated by those of ordinary skill in the art that
the accelerometer measurements obtained at 104 commonly include numerous
acceleration components. For example, depending on the drilling
conditions and the accelerometer configuration, such measurements may
include: (i) a gravitational acceleration component due to the
gravitational field of the earth, (ii) a centripetal acceleration
component due to the rotational speed of the sensor sub body, (iii) a
tangential acceleration component due to the rotational acceleration of
the sensor sub body, and (iv) one or more vibrational components due to
lateral and/or axial vibration of the drill string. Components (i), (ii),
and (iii) may be considered as unwanted noise in applications in which
the accelerometer measurements are being used as an indicator of lateral
and/or axial vibration. In certain embodiments, it may therefore be
advantageous to remove one or more of the non-vibrational components of
the accelerometer measurements. For example only, method 100 may further
optionally include the removal of any one, two, or all three of the
following: (i) a gravitational acceleration component at 114, (ii) a
tangential acceleration component at 116, and/or (iii) a centripetal
acceleration component at 118 (since these accelerations may register in
the x-, y- and/or z-axis accelerometers and be taken to be the result of
lateral vibration).

[0033] With continued reference to FIG. 4, the accelerometer measurements
made at 104 may be made at a rapid interval so as to be sensitive to
drill string vibration. The interval may be in the range from about
0.0001 to about 0.1 second (i.e., a measurement frequency in the range
from about 10 to about 10,000 Hz). For example, in one embodiment a
measurement interval of 10 milliseconds (0.01 second) may be successfully
utilized. These accelerometer measurements may then be numerically
differentiated at 106, for example, as follows:

Aid=Ai(n)-Ai(n-1) Equation 1

[0034] where Aid represents the differentiated accelerometer
measurements (i.e., a difference between sequential acceleration
measurements), Ai represents a measured acceleration value made along the
i-axis (i being representative of the x-, y-, and/or z-axis), and n
represents the array index in the set of accelerometer measurements such
that Ai(n-1) and Ai(n) represent sequential accelerometer measurements.
It will be understood that the differentiation may be performed one
measurement point at a time (i.e., as each data point is acquired) or as
a set of measurements after a predetermined number of measurements has
been acquired. The disclosed methods are not limited in these regards.

[0035] The differentiated accelerometer measurements may then be processed
to obtain a vibration parameter at 108, for example, by computing a
difference between the maximum and minimum values of the differentiated
accelerometer measurements, for example, as follows:

Ai.sub.Δ=maxAid-minAid≈2maxAid Equation 2

[0036] where Ai.sub.Δ represents the vibration parameter and max
Aid and min Aid represent the maximum and minimum
differentiated acceleration values over a predetermined time period or
for a predetermined number of samples (e.g., as determined in Equation
1). It will be understood that the differentiated accelerometer
measurements (e.g., from Equation 1) may be integrated and smoothed prior
to computing the difference in Equation 2. Such sub-sampling may enable
the vibration severity to be evaluated at substantially any suitable
frequency. In the embodiments described above, the original sampling rate
is 100 samples per second. By integration, the differentiated data may be
sub-sampled at substantially any other suitable frequency, for example,
including 10 or 50 samples per second. The sub-sampled data may then be
evaluated so as to monitor the vibration severities at predetermined
frequencies (i.e., at other measurement intervals).

[0037] In one suitable embodiment, a measurement interval of 10
milliseconds and a time period of 1 second are utilized such that the set
of acceleration differences determined in Equation 1 includes 100 raw
differentiated acceleration values. The maximum and minimum values of the
set may then be used to compute a vibration parameter in Equation 2. This
process of differentiating the accelerometer measurements over a
predetermined time period (e.g., 1 second) may then be repeated
substantially any suitable number of times to obtain a corresponding set
of vibration parameters at 110. In one embodiment, ten sequential
vibration parameters may be averaged (or summed) to obtain a single
vibration parameter which is indicative of the drill string vibration
within a 10 second time window (i.e., over 10 one-second time periods). A
smoothing algorithm may alternatively be utilized in which the vibration
parameters may be averaged (or summed) with predetermined nearest
neighbors to determine a vibration parameter which is indicative of the
drill string vibration within a one-second time window. Such smoothing
may be advantageous for computing vibration severities that may be
transmitted in real-time to the surface thereby enabling the driller to
change certain drilling parameters if necessary and to observe the
effects of such changes (e.g., to drill string rotation rate, weight on
bit, drilling fluid flow rate, etc.). The disclosed methods are not
limited in regard to such averaging or smoothing techniques. The
parameter obtained directly from Equation 2 (with no averaging or
smoothing) may likewise be utilized.

[0038] Removal of various non-vibration acceleration components may be
advantageous in certain embodiments so as to isolate the vibrational
component(s) and to obtain a corrected vibration parameter. For example,
a gravitational acceleration component may be optionally removed at 114
from the vibration parameter determined in Equation 2 as follows:

[0040] FIGS. 5A and 5B depict one methodology for determining GiA. As
depicted on FIG. 5A, the instantaneous gravitational acceleration as well
as the differentiated gravitational acceleration Gi at the sensor set is
mathematically related to the borehole inclination (Inc) and the toolface
angle of the accelerometer (θ). Following the methodology of
Equations 1 and 2, the gravitational acceleration component may be
expressed mathematically, for example, as follows:

Gi.sub.Δ=maxGid-minGid≈2maxGid Equation 4

[0041] where max Gid and min Gid represent the maximum and
minimum differentiated gravitational acceleration values. It is well
known that the maximum slope of a sine wave is located at the zero
crossing as indicated in FIG. 5B. The maximum differentiated
gravitational acceleration may be represented mathematically, for
example, as follows:

maxGid=g sin(Inc)[sin(Δθ/2)-sin(-Δθ/2)]
Equation 5

[0042] where Δθ represents the toolface angle change over the
predetermined measurement interval described above (e.g., the change in
toolface angle over a 10 millisecond interval between sequential
accelerometer measurements), g represents the gravitational acceleration
of the earth (which is well known to be approximately 9.8 m/sec2),
and Inc represents the borehole inclination. Substituting Equation 5 into
Equation 4 and recognizing that sin θ=θ for small angles and
that Δθ=2πtR/60, where t represents the predetermined
measurement interval in units of seconds and R represents the rotational
velocity of the accelerometer in units of RPM, yields:

Gi Δ = π 15 Rtg sin ( Inc ) Equation
6 ##EQU00001##

[0043] where i represents one of the cross-axial axes (i.e., x- or the
y-axis). Note that the cross-axial gravitational acceleration component
is a maximum in a horizontal well (90 degree inclination) and near zero
in a vertical well (zero degree inclination). The axial gravitational
acceleration component is described in more detail below.

[0044] As indicated in Equation 6, the gravitational acceleration
component may be removed from the vibration parameter to obtain a
corrected vibration parameter when the borehole inclination and rotation
rate of the sensor are known. As is well known in the art, the borehole
inclination may be obtained from the accelerometer measurements, for
example, according to one of the following equations:

[0045] where Ax, Ay, and Az represents the measured tri-axial acceleration
values as described above and mag(G) represents the magnitude of the
earth's gravitational field. The magnitude of the earth's gravitation
field may obtained from geological surveys, measured on site, or
determined from the accelerometer measurements, e.g., via magG= {square
root over ((Ax2+Ay2+Az2))}. The rotation rate of the
sensor sub may also be obtained from the accelerometer measurements but
is generally obtained from substantially simultaneous magnetometer
measurements, for example, as follows:

[0046] where R represents the rotation rate in units of RPM, ω
represents the angular velocity in units of radians per second,
θm represents the magnetic toolface, t represents the
predetermined measurement interval, and n represents the array index in
the set of magnetic toolface measurements such that θm(n-1)
and θm(n) represent sequential magnetic toolface measurements.
Those of ordinary skill in the art will readily appreciate that tan
θm=My/Mx where Mx and My represent the x-axis and y-axis
magnetometer measurements. Those of ordinary skill will also be readily
able to re-write Equation 8 such that the rotation rate is expressed in
alternative units such as in radians per second or degrees per second
(the disclosed embodiments are not limited in these regards). Equation 8
may also be written with respect to accelerometer based toolface
measurements in which tan θa=Ay/Ax. Moreover, gravity toolface
and magnetic toolface may be computed from one another by adding (or
subtracting) the angle difference between them, where the angle
difference may be computed, for example from a conventional static
survey.

[0047] With reference again to FIG. 4, the tangential acceleration
component may be optionally removed from the vibration parameter at 116
to obtain a corrected vibration parameter, for example, as follows:

Vi.sub.Δ=Ai.sub.Δ-Ti.sub.Δ Equation 9

[0048] where Vi.sub.Δ represents the corrected vibration parameter,
Ti.sub.Δ represents the tangential acceleration component, and
Ai.sub.Δ represents the vibration parameter described above with
respect to Equation 2. In Equations 9-14, i represents one of the
cross-axial axes (i.e., x- or the y-axis) as there is generally minimal
z-axis (axial) tangential or centripetal acceleration. Tangential
acceleration is related to the angular acceleration (i.e., the rate of
change of the rotation rate) of the sensor (the accelerometer) and may be
expressed mathematically, for example, as follows:

Ti = r α = r [ ω ( n ) - ω
( n - 1 ) t ] Equation 10 ##EQU00004##

[0049] where Ti represents a substantially instantaneous tangential
acceleration, r represents the radial distance between the accelerometer
and the center of the sensor sub (i.e., the radius), α represents
the angular acceleration of the sensor, ω represents the angular
velocity of the sensor, t represents the predetermined measurement
interval, and n represents the array index in the set of angular velocity
measurements such that ω(n-1) and ω(n) represent sequential
angular velocity measurements. The angular velocity ω may be
obtained by differentiating the magnetic toolface measurements, for
example, as shown below in Equations 13 and 19. Following the methodology
of Equations 1 and 2, a tangential acceleration component Ti.sub.Δ
may be expressed mathematically, for example, as follows:

Ti.sub.Δ=maxTi-minTi≈2maxTi Equation 11

[0050] where max Ti and min Ti represent the maximum and minimum
instantaneous tangential accelerations within a set of measurements (made
for example within a predetermined time period).

[0051] With continued reference to FIG. 4, a centripetal acceleration
component may be optionally removed from the vibration parameter at 118,
for example, as follows:

Vi.sub.Δ=Ai.sub.Δ-Ci.sub.Δ Equation 12

[0052] where Vi.sub.Δ represents the corrected vibration parameter,
Ci.sub.Δ represents the centripetal acceleration component, and
Ai.sub.Δ represents the vibration parameter as described above in
Equation 2. When utilizing an accelerometer arrangement such as that
depicted on FIG. 3, the measured centripetal acceleration tends to be
near zero, however, removal of the centripetal acceleration component may
be advantageous when utilizing alternative accelerometer arrangements.
Centripetal acceleration is related to the angular velocity (i.e., the
rotation rate) of the sensor sub and may be expressed mathematically, for
example, as follows:

[0053] where Ci represents a substantially instantaneous centripetal
acceleration, r represents the radial distance between the accelerometer
and the center of the sensor sub (i.e., the radius), ω represents
the angular velocity of the sensor, θm represents the magnetic
toolface of the sensor, t represents the predetermined measurement
interval, and n represents the array index in the set of magnetic
toolface measurements such that θm(n-1) and θm(n)
represent sequential magnetic toolface measurements. Following the
methodology of Equations 1 and 2, the centripetal acceleration component
CxA may be expressed mathematically, for example, as follows:

Ci.sub.Δ=maxCi-minCi Equation 14

[0054] where max Ci and min Ci represent the maximum and minimum
instantaneous centripetal accelerations within a set of measurements
(made for example within a predetermined time period). Those of ordinary
skill in the art will readily appreciate that Equations 8, 10, and 13 may
be equivalently expressed in terms of angular acceleration and angular
velocity vectors {right arrow over (α)} and {right arrow over
(ω)} (the disclosed embodiments are not limited in this regard).

[0055] It will be understood that tangential and centripetal accelerations
are primarily sensed by the cross-axial accelerometers (i.e., the x- and
y-axis accelerometers) while the axial accelerometer (the z-axis
accelerometer) tends to be insensitive tangential and centripetal
accelerations. However, misalignment of the accelerometers with the
previously defined tool coordinate system can result in significant
tangential and centripetal accelerations being sensed by all three
accelerometers.

[0056] It will further be understood that the vibration parameter
corrections described above with respect to Equations 3-14 may make use
of substantially simultaneous magnetic field measurements. For example,
substantially instantaneous magnetic toolface measurements may be
computed from magnetic field measurements made at the predetermined time
interval (e.g., via tan θm=My/Mx where Mx and My represent the
x-axis and y-axis magnetometer measurements). The magnetic toolface may
be differentiated as given in Equation 19 to obtain substantially
instantaneous angular velocities which may in turn be further
differentiated as shown in Equation 10 to obtain substantially
instantaneous angular accelerations. It will further be understood that
the accelerometer and magnetometer sensors commonly include hardware
low-pass filters (as described above). These filters may have different
cut-off frequencies and phase responses. In general, accelerometers have
lower cut-off frequencies as their measurements are more sensitive to
shock and vibration. Notwithstanding, such hardware filter
characteristics difference may be compensated digitally using techniques
known to those of ordinary skill in the art.

[0057] In one example of the disclosed method embodiments, a lateral
vibration parameter may be obtained via combining both cross-axial
accelerometer measurements (the x-axis and y-axis accelerometers). The
combined lateral vibration parameter may be computed, for example, as
follows:

Vxy= {square root over (Vx2+Vy2)} Equation 15

[0058] where Vxy represents the combined lateral vibration parameter and
Vx and Vy represent the cross-axial lateral vibration parameters
computed, for example, via one of Equations 2, 3, 9, or 12 using
corresponding x- and y-axis accelerometer measurements. Also, by
analyzing the sign (vibration direction) of both x-axis and y-axis
vibrations (Vxy), the type of lateral vibration (e.g. forward whirl,
backward whirl, chaotic whirl etc.) and the movement of drillstring,
stabilizer, and bit (depending on sensor position) may be identified.

[0059] In another example, an axial vibration parameter may be readily
obtained using the axial (z-axis) accelerometer, for example, via
Equation 2 or 3. The z-axis accelerometer is not generally sensitive to
tangential or centripetal accelerations as described above, and hence
removal of these components is not generally advantageous. However, it
may be advantageous to remove a gravitational acceleration component, for
example, following the procedure described above with respect to
Equations 3-6 such that:

[0060] where Vz.sub.Δ represents the corrected axial vibration
parameter, Az.sub.Δ represents the axial vibration parameter,
Gz.sub.Δ represents the axial gravitational acceleration component,
R represents the rotation rate of the sensor sub in units of RPM, t
represents the predetermined measurement interval in units of seconds, g
represents the gravitational acceleration of the earth, and Inc
represents the borehole inclination. Note that axial gravitational
acceleration component is maximum in a vertical well (zero degree
inclination) and near zero in a horizontal well (90 degree inclination).
The rotation rate of the sensor sub may be determined via simultaneous
magnetometer measurements as described above.

[0061] The previously described magnetometer measurements may also be
utilized to obtain a stick/slip parameter (a torsional vibration
parameter), thereby enabling a full suite of dynamics measurements to be
obtained using the navigational sensors (i.e., lateral vibration, axial
vibration, and torsional vibration). Stick/slip is commonly quantified in
the industry as a maximum drill string rotation rate minus a minimum
drill string rotation rate within some predetermined time period. For the
purposes of this disclosure, a stick/slip parameter may be quantified
mathematically, for example, as follows:

SSN = max ω - min ω ave
ω Equation 18 ##EQU00007##

[0062] where SSN represents a normalized stick/slip parameter, maxω
and minω represent maximum and minimum instantaneous angular
velocities during some predetermined time period, and aveo represents the
average angular velocity during the predetermine time period (e.g., 10
seconds). It will, of course, be appreciated that the stick/slip
parameter SS need not be normalized as shown in Equation 16, but may
instead be expressed simply as the difference between the maximum and
minimum instantaneous rotation rates maxω and minω. In
certain severe applications, stick/slip conditions can cause the drill
string to temporarily stop rotating (i.e., such that: minω=0). In
such applications, the stick/slip parameter is essentially equal to or
proportional to the maximum instantaneous rotation rate maxω. As
such, it will be understood that maxω may be a suitable alternative
metric for quantifying stick/slip conditions. This alternative metric may
be suitable for some drilling applications, especially since damage and
wear to the drill bit and other BHA components is commonly understood to
be related to the maximum instantaneous drill string rotation rate. The
maximum instantaneous rotation rate may be computed downhole and
transmitted to the surface where an operator may compare the value with
the surface controlled (average) rotation rate.

[0063] The instantaneous rotation rate may be determined via magnetometer
measurements, for example, as described above with respect to Equation
13. For example, the instantaneous rotation rate of the sensor sub may be
computed via differentiating magnetic toolface measurements as follows:

ω = θ m ( n ) - θ m ( n - 1 )
t Equation 19 ##EQU00008##

[0064] where ω represents the angular velocity of the sensor sub,
θm represents the magnetic toolface, t represents the
predetermined measurement interval, and n represents the array index in
the set of magnetic toolface measurements such that θm(n-1)
and θm(n) represent sequential magnetic toolface measurements.
Therefore a stick slip parameter may be obtained, for example, via (i)
rotating magnetic field sensors in the borehole, (ii) obtaining a
plurality of magnetic field measurements at a predetermined measurement
interval, (iii) processing the magnetic field measurements to obtain
corresponding magnetic toolface measurements, (iv) differentiating the
magnetic toolface measurements to obtain angular velocities, (v)
alternatively integrate the differentiated toolface values to obtain
sub-sampled angular velocities, and (vi) and processing the angular
velocities to obtain the stick/slip parameter. The alternative
integration step and sub-sampling step may enable a frequency dependence
of the torsioanl vibration to be evaluated, e.g. a high-frequency
torsional vibration severity (10-20 mS) and a low-frequency torsional
vibration severity (100 mS˜200 mS). In the embodiments described
above, the original sampling rate is 100 samples per second. By
integration, the differentiated data may be sub-sampled at substantially
any other suitable frequency, for example, including 10 or 50 samples per
second. The sub-sampled data may then be evaluated so as to monitor the
vibration severities at predetermined frequencies.

[0065] Magnetic field measurements may be further utilized to correct
accelerometer measurements for vibrational effects such that a corrected
gravity toolface angle may be computed. For example, the corrected
gravity toolface angle may be computed while drilling via: (i) rotating
magnetic field sensors and accelerometers in a borehole, (ii) obtaining a
plurality of magnetic field measurements and accelerometer measurements
at a predetermined measurement interval while rotating (or drilling),
(iii) processing the magnetic field measurements to obtain centripetal
and/or tangential acceleration components (e.g., via Equations 10 and 13
as described above), (iv) subtracting at least one of the centripetal and
tangential acceleration components from the corresponding accelerometer
measurements to obtain corrected accelerometer measurements, and (v)
utilizing the corrected accelerometer measurements to compute a corrected
gravity toolface. Such corrected gravity toolface measurements may be
utilized, for example, in LWD imaging tools.

[0066] It will be understood that the computed downhole dynamics
parameters may be stored in downhole memory for subsequent surface
analysis and/or transmitted to the surface during drilling to enable
substantially real time mitigation as required. Those of ordinary skill
will readily appreciate the potential benefits of transmitting the
dynamics parameter(s) while drilling so that corrective measures
(including changes to the drilling parameters) may be implemented if
necessary. Due to the bandwidth constraints of conventional telemetry
techniques (e.g., including mud pulse and mud siren telemetry
techniques), each of the dynamics parameters may be reduced to a two-bit
value (i.e., four levels; low, medium, high, and severe). Non-limiting
encoding examples are shown in Table 1 for axial and lateral vibration
parameters and Table 2 for a stick/slip parameter.

[0067] FIG. 6 depicts a flow chart of a method 200 for detecting and
quantifying a whirl magnitude (such as a drill bit or stabilizer whirl
magnitude) using rotating sensors. While the previous embodiments made
use of navigational sensors, method 200 may make use of substantially any
suitable sensor set in the drill string. The sensors may include, for
example, cross-axial accelerometer and magnetometer sets deployed
substantially anywhere in the drill string (e.g., just above the bit or
further up the string). The sensors are rotated at 202, for example,
while drilling the borehole (e.g., by rotating the drill string at the
surface or rotating the drill bit with a conventional mud motor). As
described above with respect to FIGS. 2 and 3, navigational sensors may
include a tri-axial accelerometer set and a tri-axial magnetometer set,
although the disclosed embodiments are not limited in this regard.

[0068] Sensor data is acquired at a predetermined time interval at 204
while rotating in 202 (e.g., during the actual drilling process) to
obtain a set (or array) of sensor measurements. The sensor measurements
may be made at a rapid interval so as to be sensitive to whirl (and other
drill string vibrational modes as described above with respect to FIG.
4). The interval may be in the range from about 0.0001 to about 0.1
second (i.e., a measurement frequency in the range from about 10 to about
10,000 Hz). For example, in one embodiment a measurement interval of 10
milliseconds (0.01 second) may be successfully utilized.

[0069] The sensor measurements are processed at 206 to compute a rotation
rate of the sensors (or a sensor sub or tool body in which the sensors
are deployed). The instantaneous rotation rate of the sensors may be
computed, for example, via differentiating magnetic toolface measurements
as described above with respect to Equation 19. Gyroscopic (e.g., using
solid state gyroscopic sensors) and accelerometer based methods for
obtaining the rotation rate may also be utilized. The rotation rate of
the sensors is then processed in combination with a predetermined blade
count (e.g., a drill bit or stabilizer blade count) to compute a bit
frequency at 208. The sensor measurements are then further processed at
210 to compute a whirl magnitude at the whirl frequency computed in 208.
A severity of the whirl magnitude may then be classified and transmitted
to the surface if so desired. The whirl magnitude and frequency may also
be stored in downhole memory.

[0070] A bit whirl frequency (also referred to in the art as a backward
whirl frequency) may be computed based on the number of blades in a PDC
drill bit, for example, as follows:

ψwhirl=ωM=ω(N+1) Equation 20

[0071] where ωwhirl represents the bit whirl frequency, ω
represents the rotation frequency (or the instantaneous rotation
frequency) of the sensor sub (or drill string), N represents the number
of blades on the PDC drill bit, and M represents the number of lobes in
the "star" whirl pattern (which is described in more detail below with
respect to FIG. 7).

[0072] Upon acquiring a whirl frequency, the whirl magnitude may be
computed from the accelerometer measurements. For example, a set of
accelerometer measurements (e.g., a set of 1000 measurements obtained in
a 10 second interval or a set of 10,000 measurements obtained in a 100
second interval) may be transformed from the time domain to the frequency
domain. Suitable transforms include a Fourier Transform, a cosine
transform, a sine transform, a polynomial transform, a Laplace transform,
a Hartley transform, a wavelet transform, and the like. A transform may
be selected, for example, in view of the ease with which it may be
handled via a downhole processor. Cosine transforms (such as the DCT) may
reduce downhole processing requirements in that they make use of only
real-number coefficients (as opposed to complex coefficients). Fast
transforms may also be utilized, for example, including a Fast Fourier
Transform (FFT) or a Fast Cosine Transform (FCT). Such transforms are
known to those of ordinary skill in the art and are commercially
available, for example, via software such as MathCad® or
Mathematica® (Wolfram Research, Inc., Champaign, Ill.), or
MATLAB® (The Mathworks Inc.). Upon obtaining the transformed data set
(the frequency domain data set), whirl may be quantified, for example,
via obtaining the signal amplitude at the computed whirl frequency or by
integrating the signal over some frequency range about the whirl
frequency (e.g., within 1 Hz of the whirl frequency).

[0073] In an alternative methodology, a digital band pass filter may be
applied to a set of accelerometer measurements (e.g., a set of 1000
measurements obtained in a 10 second interval or a set of 10,000
measurements obtained in a 100 second interval). The magnitude of the
filtered data set may then be taken to be an indication of the whirl
magnitude. The magnitude of the filtered data set may be expressed as a
root mean square (RMS), for example, as follows:

[0075] The digital band pass filter may include, for example, a filter
having a center frequency about equal to the computed whirl frequency and
a pass band of one or two Hertz about the center frequency. Suitable
filters may include, for example, digital finite impulse response (FIR)
and infinite impulse response (IIR) filters. Those of ordinary skill in
the art will readily be able to design suitable digital band pass filters
having substantially any suitable center frequency and pass band. Code
for computing such filters is available, for example, from MATLAB®
(The Mathworks Inc.).

[0076] A suitable digital filter may be computed uphole or downhole. For
example, a downhole controller may be programmed with instructions for
computing filter coefficients for a suitable digital filter based on the
computed whirl frequency. Alternatively and/or additionally numerous
filters (sets of filter coefficients) may be computed uphole and stored
in downhole memory. These filters may be computed for each of a plurality
of expected whirl frequencies within a predetermined range of
frequencies. A suitable filter may then be obtained from memory based on
the computed whirl frequency.

[0077] As with the dynamics parameters disclosed above, the whirl
magnitude may also be stored in downhole memory for subsequent surface
analysis and/or transmitted to the surface during drilling to enable
substantially real time mitigation as required. Due to the aforementioned
bandwidth constraints of conventional telemetry techniques, the whirl
magnitude may be reduced, for example, to a one or two-bit value. A one
bit value may indicate whether or not the magnitude exceeds some
predetermined threshold. Two bits may indicate four magnitude levels,
e.g., low, medium, high, and severe. The numerical value of the whirl
magnitude may also be transmitted to the surface, e.g., as eight-bit or
sixteen-bit floating point values. The whirl magnitude may also be
normalized, for example, with respect to drill bit diameter or borehole
inclination. Moreover, the whirl magnitude may also be combined with
other dynamics measurements (e.g., the axial and lateral vibration and
stick slip measurements described above) so as to obtain a combined
parameter (e.g., a whirl/stick slip parameter, and so on).

[0078] A transmitted whirl magnitude (e.g., a bit whirl magnitude) may be
used in an automated drilling routine. For example, the whirl magnitude
and other dynamics parameters (such as stick slip) may be processed at
the surface and used to automatically adjust one or more drilling
parameters. These drilling parameters may include, for example, weight on
bit (WOB), drill string rotation rate (RPM), and drilling fluid flow
rate. In one automated embodiment a whirl magnitude received at the
surface may be compared with a predetermined threshold. If the whirl
magnitude is greater than the threshold and the WOB is less than some
maximum value, then the WOB may be incrementally increased (e.g., by
about 10%). If the whirl magnitude remains greater than the threshold and
the RPM is greater than some minimum value, then the RPM may be
incrementally decreased (e.g., by 10%). The routine may be continually
repeated as long as desired. The routine may further process received
stick slip values. For example, if the received stick slip is greater
than a threshold and the WOB is greater than some minimum value, then the
WOB may be incrementally decreased (e.g., by about 5%). If the stick slip
remains greater than the threshold and the RPM is less than some maximum
value, the RPM may then be incrementally increased (e.g., by 10%). By
using these routines in combination the drilling parameters may be
automatically maintained within a range of values that tends to minimize
both whirl and stick slip.

[0079] It will of course be understood that the raw magnetometer and
accelerometer data may be transmitted to the surface (e.g., using a wired
drillpipe) and that the raw data may be processed at the surface
according to any one or more of the various methods disclosed herein.

[0080] The various disclosed embodiments are now described in further
detail by way of the following example, which is intended to be an
example only and should not be construed as in any way limiting the scope
of the claims. Sensor data was obtained using the methodology described
above in a section of a borehole that was being drilling in a shale
formation. The navigational sensors were deployed in a PathFinder®
iPZIG® sensor sub deployed immediately above the bit that included
conventional tri-axial accelerometers and tri-axial flux-gate
magnetometers. The accelerometer configuration was similar to that
depicted on FIG. 3. A conventional mud motor (having a bent housing) was
deployed above the iPZIG® sensor sub. A conventional EM
(electromagnetic) short-hop enabled two-way communication with other
tools in a BHA (such as MWD and telemetry tools) across the motor. It
will of course be understood that the disclosed embodiments are not
limited to the use of a near-bit sensor sub, but are equally applicable
to the MWD directional module deployed further away from the bit and/or
other LWD imaging tools (gamma, density, neutron, caliper, resistivity
imaging tools) including a directional sensor package.

[0081] FIGS. 7 and 8 depict plots of accelerometer and magnetometer data
obtained during the aforementioned drilling operation. FIGS. 7A, 7B, 7C,
and 7D (collectively FIG. 7) depict plots of the x- and y-axis
accelerometer and magnetometer data obtained during the drilling
operation. It will be understood that the x- and y-axes are transverse to
the longitudinal axis of the drill string (also referred to as
cross-axial) as described above with respect to FIGS. 2 and 3. FIG. 7A
depicts a plot of the y-axis versus the x-axis accelerometer (AY vs.
AX) values for each of 1000 data points acquired in a 10 second
interval. The lobed "star" pattern indicative of bit whirl is readily
apparent. In this particular example, the PDC drill bit included six
blades and thus the star pattern includes seven lobes. A pattern
recognition and/or image processing algorithm may be applied to the
AY vs. AX plot in order facilitate bit whirl identification.
FIG. 7C depicts a plot of the y-axis versus the x-axis magnetometer
(BY vs. BX) values. The circular pattern is indicative of
rotary motion.

[0082] FIGS. 7B and 7D depict frequency spectra of plots of AX (FIG.
7B) and BX (FIG. 7D) from which the DC components have been removed.
The peak in both spectra show the drill bit rotation frequency, which was
about 2.3 Hz (138 rpm). It will thus be understood that obtaining a
frequency spectrum of the accelerometer or magnetometer data is an
alternative means for obtaining the rotation rate of the sensor sub at
206 of method 200 (FIG. 6).

[0084] FIGS. 8A and 8C depict plots of AXY and BXY versus time
for the 10 second interval over which the sensor measurement were
acquired. FIGS. 8B and 8D depict frequency spectra for the time domain
plots of and BXY shown FIGS. 8A and 8C. The DC component of the and
BXY spectra has been removed. The spectrum depicted on FIG. 8B shows
a clear and strong peak at the bit whirl frequency of 16.2 Hz (the
rotation frequency 2.3 Hz times the 7 lobes). The BXY spectrum
depicted on FIG. 8D shows a peak at 4.6 Hz (the second harmonic of the
2.3 Hz rotation frequency). The small peak at the fundamental rotation
frequency (2.3 Hz) may be taken to be indicative of the drill string
rotation being smooth (nearly constant) in the 10 second interval over
which the sensor data was acquired. It will be understood that since AC
signals are used to detect the whirl frequency, AC coupled accelerometers
and/or magnetometers may be used in whirl detection.

[0085] While the example described above makes use of x- and y-axis
accelerometer measurements to compute AXY, it will be understood
that the disclosed embodiments are so limited. In an alternative
embodiment a single radially oriented accelerometer to directly measure
AXY. The AXY spectrum may then be obtained by transforming the
radial sensor data into the frequency domain as described above.

[0086] It will be understood that while not shown in FIGS. 1, 2, and 3,
bottom hole assemblies suitable for use the disclosed embodiments
generally include at least one electronic controller. Such a controller
may include signal processing circuitry including a digital processor (a
microprocessor), an analog to digital converter, and processor readable
memory. The controller may also include processor-readable or
computer-readable program code embodying logic, including instructions
for computing vibrational parameters as described above, for example, in
Equations 1-19. One skilled in the art will also readily recognize that
the above mentioned equations may also be solved using hardware
mechanisms (e.g., including analog or digital circuits).

[0087] A suitable controller may further include a timer including, for
example, an incrementing counter, a decrementing time-out counter, or a
real-time clock. The controller may further include multiple data storage
devices, various sensors, other controllable components, a power supply,
and the like. The controller may also optionally communicate with other
instruments in the drill string, such as telemetry systems that
communicate with the surface or an EM (electro-magnetic) shorthop that
enables the two-way communication across a downhole motor. It will be
appreciated that the controller is not necessarily located in the sensor
sub (e.g., sub 60), but may be disposed elsewhere in the drill string in
electronic communication therewith. Moreover, one skilled in the art will
readily recognize that the multiple functions described above may be
distributed among a number of electronic devices (controllers).

[0088] Although downhole dynamics measurements using navigational sensors
and certain advantages thereof have been described in detail, it should
be understood that various changes, substitutions and alternations can be
made herein without departing from the spirit and scope of the disclosure
as defined by the appended claims.